
\(\mathbb {U}\)  is a proper filter.

\(\lnot \exists \mathbb {W}(\mathbb {U}\subset  is a filter).\) 

Here we have to pay a little more attention, however. Unlike set-theoretic ultrafilters, which always exist for any nonempty set \(S\)  as subsets of \(\mathcal {P}(S)\) , for a portion of partial orders or preorders there exist no ultrafilters over them. To give a pictorial though informal depiction, those partial orders without ultrafilters never “diverge permanently” towards totally independent directions, thus so to speak, they are either just linear orders or in a sense much like linear orders. Fortunately, such kind of partial orders matter little to argumentation theory — no need to argue if there is no divergence after all! Therefore, as the following Definition REF , in this paper we can safely concentrate on nontrivial argument Kripke models with ultrafilters.
[Triviality]
An argument Kripke model \(\mathfrak {N}\)  is trivial iff there is no ultrafilter over the preorder \(\le \)  in Definition . \(\mathfrak {N}\)  is nontrivial iff it is not trivial.

Through viewing ultrafilters as possible worlds, we at last approach the following Definition REF :
[Generated Epistemic Kripke Model]
Given a nontrivial argument Kripke model \(\mathfrak {N}=(\mathbb {W},A,G)\)  with a possible argument \(W\in \mathbb {W}\) , the generated epistemic Kripke model is \(\mathfrak {M}_g=(S,E,F)\) , where:

\(S=\lbrace \mathbb {U}\subseteq \mathbb {W}\mid \mathbb {U}\)  is an ultrafilter over the preorder \(\le \rbrace \) .

For any \(i\in I\)  and any \(\mathbb {U},S\) , \((\mathbb {U},\in E(i)\)  iff \(\mathbb {U}\cap G(i)=G(i)\) .

For any \(p\in P\) , \(F(p)=\lbrace \mathbb {U}\in S\mid \exists U\in \mathbb {U}\forall V\in \mathbb {U},V\le U\rightarrow (V,W)\notin A(p)\rbrace \) .

It is easy to see that \(S\ne \emptyset \)  and that \(E(i)\)  is an equivalence relation, thus \(\mathfrak {M}_g\)  in Definition REF  is indeed a well-defined epistemic Kripke model. Readers may now wonder, however, that we have yet provided no justification for \(E(i)\)  or \(F(p)\)  in Definition REF . As a matter of fact, since possible worlds are constituted of ultrafilters, which are somewhat mathematically factitious after all, it is inevitably tough to give further intuitive explanation solely based on our folk experience. Rather, we shall adopt a little indirect but strictly formal approach, through investigating into a beautiful duality theorem on invariance over the language in Definition , so as to demonstrate that our generations in Definitions  and REF  are indeed perfectly reasonable. Nevertheless, before we dive into technical subtleties related to this duality theorem in Section , we had better first take a look at one vivid practical example presented in the following Section , both for acquainting readers with our rather abstract mathematical definitions above and for illustrating the significant practicality of such two-way generation as well as our framework as a whole.
A Practical Example: Running for the Mayor
Alice and Bob, two of the mayor candidates, are publicly debating over their intended urban policies.
`Aging is the biggest challenge our city is now facing,' says Alice. `There are more and more old people, yet young population is decreasing. If I am elected as the mayor, I will wield a series of revolutionary measures that attract young people to move and dwell into our city.'\(\cdots \cdots (A_1)\)
`No, we don't need any more young people who will end up competing for only a limited number of job positions,' says Bob. `In my opinion, it is unemployment that constitutes our biggest problem, since at present around 1,000 young people are badly in need of a job. Therefore if I am elected as the mayor, I promise to create 1,000 new job opportunities during 5 years.'\(\cdots \cdots (B)\)
`I cannot agree with you,' says Alice. `As far as I know, our city's working-age population is rapidly decreasing at a current speed of around minus 400 per year, so what will be the point of those 1,000 more job opportunities if there are merely not enough young people who can take them?'\(\cdots \cdots (A_2)\) 

In actuality, the above debate depicted in Example  may well continue going on, but to utter a real-time commentary on the debate, at the present moment Alice's last argument attacks Bob's, and thus based on our folk principle that “the one who has the last word laughs best”, we can conclude that Alice is tentatively winning the debate. Therefore, we can formalize this debate as the following:

Two agents \(a,b\)  denote Alice and Bob, respectively.

\(A_1\) , \(B\) , \(A_2\)  denote the three consecutive arguments by Alice and Bob, respectively, as we have indexed beside the text above in Example . Our current possible argument is the last argument, namely \(A_2\) .

Proposition \(p\)  denotes that “there exists considerable unemployment.” Argument \(B\)  attacks argument \(A_1\)  with respect to proposition \(p\) .

Proposition \(q\)  denotes that “considering aging, unemployment is not a serious problem”. Argument \(A_2\)  attacks argument \(B\)  with respect to proposition \(q\) .

At present, arguments \(A_1\)  and \(A_2\)  are of course available to Alice herself. Nonetheless, argument \(B\)  is not available to Bob now (although it used to be), because it is losing to the current argument \(A_2\)  which is attacking it and hence Bob has to give it up.

Then, our current argument Kripke model \(\mathfrak {N}\)  should look like the following:
\(@=50pt{\mathfrak {N}& A_1:a,\lnot b[r]|{A(p)} & B:\lnot a,\lnot b[r]|{A(q)} & A_2:a,\lnot b}\) 
Based on model \(\mathfrak {N}\) , the preorder \(\le \)  in Definition  can be represented as the following Hasse diagram, namely \(A_2\le A_1\)  and \(B\le A_1\) :
\({& A_1 &\\A_2@{-}[ur] & & B@{-}[ul]}\) 
And so obviously there exist two ultrafilters, namely \(\lbrace A_1,A_2\rbrace \)  and \(\lbrace A_1,B\rbrace \) .
Before finalizing our construction of the generated epistemic Kripke model \(\mathfrak {M}_g\) , let us pause for a while and genuinely think about what we are actually doing now and why we are doing it. Surely, we desire a familiar, tidy and expressive epistemic Kripke model that can aid us to reason about the current scenario, as epistemic logic has been thoroughly studied over the decades and has proved to be a quite accurate and productive depiction of agents' knowledge, especially helpful when dealing with intricate situations such as the muddy children puzzle and the Byzantine generals problem [1]}. Nonetheless, there remains one fatal concern. Indeed, the common method to obtain epistemic Kripke models in reality still relies largely on human intuition and so is rather obscure, a deficiency which in fact greatly impedes such kind of logic approach to knowledge presentation from being universally put into practical application [2]}. For instance, just under our current setting of the above Example , we have already formalized the argument Kripke model \(\mathfrak {N}\)  with ease, simply as a literal transcription of the debate per se, however, the task of generating the epistemic Kripke model for Alice and Bob is far from obvious or even natural. After all, in the realistically physical world, it is solely Alice and Bob's extrinsic debate that is plainly accessible to other audience, whereas their intrinsic knowledge could be indirectly inferred but never directly observed.
The above analysis accounts for exactly why we are about to compute the generated epistemic Kripke model \(\mathfrak {M}_g\)  from our original argument Kripke model \(\mathfrak {N}\) , and also illuminates one of the major motivations as well as applications of our dual framework presented in this paper. Hence as defined in the previous Section , the generated epistemic Kripke model \(\mathfrak {M}_g\)  from the argument Kripke model \(\mathfrak {N}\)  together with the current possible argument \(A_2\)  is calculated to be the following:
\(@=50pt{\mathfrak {M}_g & \lbrace A_1,A_2\rbrace :p,q@(ur,ul)@{.}|{E(a),E(b)}@{.}[r]|{E(b)} & \lbrace A_1,B\rbrace :p,\lnot q@(ur,ul)@{.}|{E(a),E(b)}}\) 
It can be noted that \(\mathfrak {M}_g\)  contains two possible worlds, namely the two ultrafilters \(\lbrace A_1,A_2\rbrace \)  and \(\lbrace A_1,B\rbrace \) . Thus which one should be our current possible world? The current possible world should of course include the current possible argument \(A_2\) , and so it must be the ultrafilter \(\lbrace A_1,A_2\rbrace \) , where both \(p\)  and \(q\)  really hold. Now as \(\mathfrak {M}_g\)  also shows, Alice actually knows the fact \(q\)  but Bob does not, since he also considers another world (i.e. ultrafilter) \(\lbrace A_1,B\rbrace \)  possible, where \(\lnot q\)  holds instead. Hence in a word, our resulting epistemic Kripke model \(\mathfrak {M}_g\) , albeit automatically generated from the argument Kripke model \(\mathfrak {N}\) , indeed conforms pretty well with what a human being would intuitively deduce from the argumentation process between Alice and Bob in Example .
Admittedly, in this specific simple example, \(\mathfrak {M}_g\)  looks so obvious just by our intuition, and some of the readers might still be wondering why we essentially need such an indirect process to generate \(\mathfrak {M}_g\)  rather than try to directly construct the current epistemic Kripke model. But here the main point is, no matter how simple or how complex \(\mathfrak {M}_g\)  may actually be, to say the very least, any straightforward clues about what the epistemic Kripke model is like can be scarcely found either in the real physical world, or in the narration of some story happened in real world. For instance considering the muddy children puzzle, an ordinary solution would commonly start with a cubic epistemic Kripke model which is almost taken for granted, although this puzzle's literal text — namely some English sentences describing the background settings of the puzzle followed by the conversation between the parent and the children — actually shows no direct hint for any cubic structure at all. As a matter of fact, according to David Lewis' modal realism [3]}, all possible worlds in the epistemic Kripke model are equally real, which could suggest that since we are at our current world, we should not be fully capable of observing other possible worlds as directly as the current world, thus not for the whole epistemic Kripke model, either. Therefore in argumentation theory as well as other fields of application, quite substantially, the intangible epistemic Kripke model is always generated from other concrete observables, such as the argument Kripke model. Even though in relatively easy cases like Example  and the muddy children puzzle, the epistemic Kripke model seems fairly apparent, nonetheless, human's intuition may fail to function well when confronted with much complicated circumstances, and moreover importantly, such vague concept of human intuition possesses little reference value for AI systems to be counted as an effective algorithm. Hence, our approach by the dual theory helps to fill up the above gap through automatically generating the appropriate epistemic Kripke model in a systematical way.
As a conclusion to our philosophical analysis over Example  in this section, we have carefully demonstrated how to generate the corresponding epistemic Kripke model \(\mathfrak {M}_g\)  given the current argument Kripke model \(\mathfrak {N}\) . In the next Section , we shall then turn our attention to some technical details and in the end prove a duality theorem so as to formally justify our such methodology of two-way generation.

Knowledge vs. Argumentation: The Duality
Fix an epistemic Kripke model \(\mathfrak {M}=(S,E,F)\)  with a possible world \(s\in S\) , and suppose its corresponding generated argument Kripke model is \(\mathfrak {N}_g=(\mathbb {W},A,G)\) , where we have \(\lbrace s\rbrace \in \mathbb {W}\) . Assume \(\mathfrak {N}_g\)  is nontrivial. Thus once again, with respect to this fixed argument Kripke model \(\mathfrak {N}_g\)  and this fixed possible argument \(\lbrace s\rbrace \) , suppose its corresponding generated epistemic Kripke model is \(\mathfrak {M}_{gg}=(S^{\prime },E^{\prime },F^{\prime })\) . We reasonably expect that \(\mathfrak {M}_{gg}\) , even though may not be exactly identical to the original \(\mathfrak {M}\) , must resemble \(\mathfrak {M}\)  to a certain extent. Nonetheless, in order to pin down such intuition precisely, we firstly have to determine which possible world in \(S^{\prime }\)  should correspond to the original \(s\in S\) , namely, represent the currently real world.

For any \(t\in S\)  namely \(\lbrace t\rbrace \in \mathbb {W}\) , there does not exist \(U\in \mathbb {W}\)  such that \(U\le \lbrace t\rbrace \wedge \lbrace t\rbrace \nleq U\) , where relation \(\le \)  is defined in Definition .


To a contradiction suppose such \(U\)  exists, since \(\lbrace t\rbrace \nleq U\) , there exists \(p\in P\) , \(V\in \mathbb {W}\)  such that \((V,U)\in A(p)\wedge (V,\lbrace t\rbrace )\notin A(p)\) , hence there does not exist \(v\in V\)  such that \((v\in F(p)\wedge t\notin F(p))\vee (v\notin F(p)\wedge t\in F(p))\) , namely for any \(v\in V\) , \(v\in F(p)\iff t\in F(p)\) . And then as \((V,U)\in A(p)\) , it is easy to see that for any \(u\in U\) , \(v\in V\) , \(u\in F(p)\iff v\notin F(p)\iff t\notin F(p)\) , thus \((U,\lbrace t\rbrace )\in A(p)\)  but of course \((U,U)\notin A(p)\) , therefore \(U\nleq \lbrace t\rbrace \) , a contradiction.


For any \(t\in S\) , \(\tau (t)=\lbrace U\in \mathbb {W}\mid \lbrace t\rbrace \le U\rbrace \subseteq \mathbb {W}\)  is an ultrafilter.


Since \(\lbrace t\rbrace \le \lbrace t\rbrace \) , \(\lbrace t\rbrace \in \tau (t)\)  so \(\tau (t)\ne \emptyset \) . For any \(U\in \tau (t)\) , any \(W\in \mathbb {W}\) , if \(U\le W\)  then by transitivity \(\lbrace t\rbrace \le W\) , hence \(W\in \tau (t)\) . For any \(U,V\in \tau (t)\) , \(\exists \lbrace t\rbrace \in \tau (t)\)  such that \(\lbrace t\rbrace \le U\wedge \lbrace t\rbrace \le V\) . Thus \(\tau (t)\)  is a filter. To a contradiction suppose \(\tau (t)=\mathbb {W}\) , then any proper filter \(\mathbb {U}\subset \mathbb {W}\)  is not an ultrafilter, namely there are no ultrafilters, contradicting our assumption that \(\mathfrak {N}_g\)  is nontrivial. Thus \(\tau (t)\subset \mathbb {W}\)  is a proper filter. To a contradiction suppose there exists \(\mathbb {U}\subseteq \mathbb {W}\)  such that \(\tau (t)\subset \mathbb {U}\)  and that \(\mathbb {U}\)  is a filter, then there exists \(U\in \mathbb {U}\setminus \tau (t)\) , as \(\mathbb {U}\)  is a filter, there exists \(V\in \mathbb {U}\)  such that \(V\le U\wedge V\le \lbrace t\rbrace \) , and because \(U\notin \tau (t)\) , we also have \(V\notin \tau (t)\) , namely \(\lbrace t\rbrace \nleq V\) , contradicting the previous Lemma . Thus \(\tau (t)\)  is an ultrafilter.

[Principal Ultrafilter]
As Proposition  shows, for any \(t\in S\) , we call \(\tau (t)\in S^{\prime }\)  as \(t\) 's principal ultrafilter.

Although the definition here is slightly different from ordinary principal ultrafilters over a Boolean algebra [4]}, readers should feel convinced that the core idea is the same and so we are quite justified for usage of the same name. Then of course, the current possible world in the “generated generated” epistemic Kripke model \(\mathfrak {M}_{gg}\)  should be \(\tau (s)\in S^{\prime }\) , namely, the principal ultrafilter of the original current world \(s\in S\) .
A little more preparation is required: we additionally assume that for any proposition \(p\in P\) , \(\mathfrak {M},s\vDash p\)  i.e. \(s\in F(p)\) . The reason is that an argument Kripke model can only inform us about the relative attack relations between arguments with respect to some proposition \(p\) , but nothing concerned with their absolute true-hood or falsehood, hence such kind of assumption, or to be called more accurately as convention or protocol, is necessarily needed. Anyhow, this restriction actually does not matter a lot, since due to symmetry between true and false we can always freely choose either \(p\)  or \(\lnot p\)  as the proper basic proposition so as to meet the assumption's requirements. We then firstly prove a series of useful lemmata under our current assumption:

For any \(U,V\in \mathbb {W}\)  such that \(U\subseteq V\) , \(U\le V\) .


This fact can be verified straightforwardly from definition.


For any \(t\in S\)  and any \(p\in P\) , \(\mathfrak {M}_{gg},\tau (t)\vDash p\iff \mathfrak {M},t\vDash p\) .


We have
\(\mathfrak {M}_{gg},\tau (t)\vDash p\iff & \tau (t)\in F^{\prime }(p)\\\iff & \exists U\in \tau (t)\forall V\in \tau (t),V\le U\rightarrow (V,\lbrace s\rbrace )\notin A(p)\\\iff & \exists U\in \tau (t)\forall V\in \tau (t),V\le U\rightarrow \lnot \exists r\in V\forall s\in \lbrace s\rbrace ,\\& (r\in F(p)\wedge s\notin F(p))\vee (r\notin F(p)\wedge s\in F(p))\\\iff & \exists U\in \tau (t)\forall V\in \tau (t),V\le U\rightarrow \forall r\in V,\\& r\in F(p)\leftrightarrow s\in F(p)\\\iff & \exists U\in \tau (t)\forall V\in \tau (t),V\le U\rightarrow \forall r\in V,r\in F(p) & (\dag )\) 

On the one hand, suppose (†) holds, then since \(U\in \tau (t)\) , we have \(\lbrace t\rbrace \le U\) , so \(\forall r\in \lbrace t\rbrace ,r\in F(p)\) , hence \(t\in F(p)\) . On the other hand, suppose \(t\in F(p)\) , then \(\lbrace t\rbrace \in \tau (t)\) , and for any \(V\in \tau (t)\)  such that \(V\le \lbrace t\rbrace \) , namely for any \(V\in \tau (t)\)  such that \(\forall q\in P\forall W\in \mathbb {W}\) , \((W,\lbrace t\rbrace )\in A(q)\rightarrow (W,V)\in A(q)\) , we thus have \((W,\lbrace t\rbrace )\in A(p)\rightarrow (W,V)\in A(p)\) . If \(\lnot \exists u\in S\)  such that \(u\notin F(p)\) , then of course \(\forall r\in V\subseteq S\) , \(r\in F(p)\) . If \(\exists u\in S\)  such that \(u\notin F(p)\) , then \((\lbrace u\rbrace ,\lbrace t\rbrace )\in A(p)\)  thus \((\lbrace u\rbrace ,V)\in A(p)\) , therefore we still have \(\forall r\in V\) , \(r\in F(p)\) . Anyway, (†) always holds. In total, we have \((\dag )\iff t\in F(p)\) , namely \(\mathfrak {M}_{gg},\tau (t)\vDash p\iff \mathfrak {M},t\vDash p\) .


For any \(t\in S\)  and any \(i\in I\) , if \((s,t)\in E(i)\)  then \((\tau (s),\tau (t))\in E^{\prime }(i)\) .


For any \(U\in \tau (s)\cap G(i)\)  we have \(t\in U\) , hence \(\lbrace t\rbrace \le U\) , so also \(U\in \tau (t)\cap G(i)\) , and vice versa, thus \(\tau (s)\cap G(i)=\tau (t)\cap G(i)\) , namely \((\tau (s),\tau (t))\in E^{\prime }(i)\) .


For any \(i\in I\) , \(p\in P\)  and any \(\mathbb {U}\in S^{\prime }\)  such that \((\tau (s),\mathbb {U})\in E^{\prime }(i)\) ,

if \(\mathfrak {M}_{gg},\mathbb {U}\vDash p\)  then there exists \(t\in S\)  such that \((s,t)\in E(i)\)  and that \(\mathfrak {M},t\vDash p\) .

if \(\mathfrak {M}_{gg},\mathbb {U}\nvDash p\)  then there exists \(t\in S\)  such that \((s,t)\in E(i)\)  and that \(\mathfrak {M},t\nvDash p\) .

Denote \([s]_i=\lbrace t\in S\mid (s,t)\in E(i)\rbrace \in \mathbb {W}\) . Since \(s\in [s]_i\) , \(\lbrace s\rbrace \le [s]_i\)  namely \([s]_i\in \tau (s)\) , also since \([s]_i\in G(i)\) , we have \([s]_i\in \tau (s)\cap G(i)\) . As \((\tau (s),\mathbb {U})\in E^{\prime }(i)\) , namely \(\tau (s)\cap G(i)=\mathbb {U}\cap G(i)\) , hence \([s]_i\in \mathbb {U}\) .
Moreover the same as (†) in Lemma , we have \(\mathfrak {M}_{gg},\mathbb {U}\vDash p\iff \exists U\in \mathbb {U}\forall V\in \mathbb {U},V\le U\rightarrow \forall r\in V,r\in F(p)\) .

If \(\mathfrak {M}_{gg},\mathbb {U}\vDash p\) , namely \(\exists U\in \mathbb {U}\forall V\in \mathbb {U},V\le U\rightarrow \forall r\in V,r\in F(p)\) , then as \([s]_i\in \mathbb {U}\)  and \(\mathbb {U}\)  is a filter, there exists \(W\in \mathbb {U}\)  such that \(W\le [s]_i\wedge W\le U\) , hence \(\forall r\in W\) , \(r\in F(p)\) , and because \(W\ne \emptyset \)  we have \(\exists r\in W\) , \(r\in F(p)\) . Since \(W\le [s]_i\)  and \((W,W)\notin A(p)\) , we have \((W,[s]_i)\notin A(p)\) , therefore \(\exists t\in [s]_i\)  such that \(t\in F(p)\) , namely \((s,t)\in E(i)\)  and \(\mathfrak {M},t\vDash p\) .

If \(\mathfrak {M}_{gg},\mathbb {U}\nvDash p\) , namely \(\forall U\in \mathbb {U}\exists V\in \mathbb {U},V\le U\wedge \exists r\in V,r\notin F(p)\) , then as \([s]_i\in \mathbb {U}\) , there exists \(V\in \mathbb {U}\)  such that \(V\le [s]_i\wedge \exists r\in V\) , \(r\notin F(p)\) . Since \(V\le [s]_i\)  and \((V,V)\notin A(p)\) , we have \((V,[s]_i)\notin A(p)\) , therefore \(\exists t\in [s]_i\)  such that \(t\notin F(p)\) , namely \((s,t)\in E(i)\)  and \(\mathfrak {M},t\nvDash p\) .

Finally we state our main result as the following duality theorem:
[Duality Theorem]
For any epistemic formula \(\varphi \)  where epistemic modalities only act upon atomic propositions and their negations, we have \(\mathfrak {M},s\vDash \varphi \iff \mathfrak {M}_{gg},\tau (s)\vDash \varphi \) .


The atomic case when \(\varphi \)  is the form of \(p\)  follows from Lemma . The modal case when \(\varphi \)  is the form of \(\mathcal {K}_i p\)  or \(\mathcal {K}_i\lnot p\)  follows from Lemmata  and  for one direction, as well as Lemma  for another direction.

It may seem a little pity that according to Theorem , only agents' knowledge about atomic propositions (and their negations) is ensured to be preserved from \(\mathfrak {M}\)  to \(\mathfrak {M}_{gg}\) , but not necessarily for more complex knowledge like \(\mathcal {K}_i(p\rightarrow q)\)  or mutual knowledge like \(\mathcal {K}_i\mathcal {K}_j p\) . However, such limitation is exactly due to the fact that in Definition  of argument Kripke models, possible arguments can attack each other only with respect to each single atomic proposition, but not with respect to a set of propositions like \(\lbrace p,q\rbrace \)  or agents' knowledge like \(\mathcal {K}_jp\) . For simplicity and clarity, we choose to work with this restricted version of definition in the major part of this paper, but shall also briefly discuss possible directions to expand the model and semantics in the following Subsection REF  so that we can obtain richer representation and finer-grained resolution.
On the other hand symmetrically, it is also totally feasible to start from an argument Kripke model \(\mathfrak {N}=(\mathbb {W},A,G)\)  with a possible argument \(W\in \mathbb {W}\)  and end up with the “generated generated” argument Kripke model \(\mathfrak {N}_{gg}\) , while a corresponding duality theorem similar to Theorem  should then be expected under a suitable set of presumptions. Anyhow, this dual generating process from \(\mathfrak {N}\)  to \(\mathfrak {N}_{gg}\)  might seem less intuitive or practically useful, and we shall omit detailed analysis here since by duality, the mathematical technique is just very similar.
In sum, this kind of duality theorem formally justifies our two-way generation in Definitions  and REF , so that the static, intrinsic epistemic Kripke models and the dynamic, extrinsic argument Kripke models are indeed a dual pair.

Further Discussion
This section collects a bunch of miscellaneous discussion on further expansions and applications of our framework presented in this paper.
Syntax Expanded
In Definition , a symmetric pair of languages, namely the epistemic formula \(\varphi \)  and the argument formula \(\xi \) , are defined as the ordinary basic modal logic. While epistemic logic has been well established in the literature, readers may keep wondering about the practical usefulness of the argument formulae, in spite of our explanation for their intuitive readings right after Definition . As a matter of fact, argument formulae are just as expressive as epistemic formulae, both mathematically and pragmatically, that is to say, they are indeed able to talk about lots of interesting features of argumentation. As an elementary example, the following very simple argument formula \(\mathcal {A}_p\lnot i\)  intuitively says that, all arguments that attack the current argument with respect to proposition \(p\)  are not accessible to agent \(i\) , in other words, the current argument “defeats”, “dominates” or “overrules” agent \(i\)  with respect to proposition \(p\)  because he has no available counter arguments to refute it, hence, whoever utters this argument out will certainly win the debate over proposition \(p\)  against agent \(i\) . Although we admit that the usage itself might seem a little bit awkward — probably because it somewhat differs from our familiar SVO word order in English — substantially, argument formulae are competently powerful for expressing a variety of situations in argumentation.
Moreover, we mainly restrict our attention in this paper onto basic modal logic for the reason of simplicity and clarity, but of course we by no means have to take such limit rigidly. Since sole argument formulae are definitely not enough for distinguishing every detail of argument Kripke models, we are rather free to expand the language, for example by introducing new modalities, so as to obtain more abundant expressivity just in any case of need.

Semantics Expanded
On the other hand, it is also plausible to consider expanding the Kripke models, especially the argument one, with more affluent structure and information. As Definition  shows, for simplicity at present, we only allow arguments to attack each other with respect to certain atomic proposition. However let alone in mathematics, even in real life, people often argue for different opinions which do not disagree simply on one atomic proposition. For example, two agents do not know whether Alice loves Bob or Bob loves Alice, hence there is no divergence on either atomic proposition, but one agent believes that Alice loves Bob if and only if Bob loves Alice, while the other agent does not think it is necessarily so. Furthermore, arguments can also disagree over agents' knowledge, such as \(\mathcal {K}_jp\) . Through taking various kinds of attack relations into consideration we will then be able to drop the restrictions on language in Theorem , as we have remarked right after that theorem's proof.
For another common concern on the relationship between our dual framework presented in this paper and Dung's argumentation theory [5]}, as we have foretold in Section , Dung's arguments seem to attack each other only in the weak sense, because accordingly in his definition of consistency, any two arguments in a consistent set can never attack one another, which seems to be even stricter than our later definition in Section  of maximal consistent sets of arguments as ultrafilters over a preorder. Anyway, although Dung's definition is perhaps not strong enough to fulfill our demand here as a dual theory, we might still consider using it in certain due case.
On the flip side, not only attack relations, but support relations between arguments may also get dealt with as another sort of modality. In total, the same technique applies to all these semantic expansions with various kinds of relations amongst the arguments.

Application to Game Theory
We would also like to suggest that our framework is versatile and flexible for applying to game theory, which contributes as another quite important practical usage. For a very rough instance, agents who are game players might take turns to pick one possible argument that is currently available to himself from the argument Kripke model, and as explained in the previous Subsection REF , agent \(i\)  will lose this game of argumentation over proposition \(p\)  against any argument on which \(\mathcal {A}_p\lnot i\)  holds. Briefly speaking, in accordance with different kinds of practical scenarios, our detailed game rules can also be set differently, and moreover, agents can devise their personal game strategies based on their own knowledge from the epistemic Kripke model.

Dynamic Essence of Knowledge as Maxwell's Demon
Commonly, knowledge is regarded as some static state of the objective world, for example, we talk about the agents' knowledge at a single moment. The traditionally accepted definition of knowledge as justified true belief or something else alike does not seem to reveal any temporal characteristic of knowledge, either [6]}. Nonetheless, our approach in this paper to treat knowledge and argumentation as a dual creatively provides an alternative angle of philosophical view, asserting that knowledge also bears an essential dynamic dimension, as the current epistemic Kripke model is always accompanied by its corresponding generated argument Kripke model.
As for an intuitive explanation, here we would like to resort to Maxwell's demon. Based on both thermodynamics [7]} and information theory [8]}, the recognized explication for how the demon works is that the demon has to know each molecule's velocity in order to perform his task, while such knowledge (or to be called information) itself is negative entropy, in other words in the physical reality, entropy must increase when the demon attempts to acquire new knowledge about a molecule's velocity by whatever means. Hence the demon's seemingly subjective knowledge as a static mental state actually carries physically objective negative entropy, exactly because the demon is able to act dynamically according to his knowledge and thus influence the physical world. Therefore, this well known thought experiment strongly defenses our metaphysical and epistemological thesis that knowledge is essentially dynamic.

Concluding Remarks
In this paper, we come up with a dual theory between static knowledge and dynamic argumentation so as to uncover the two concepts' profound relevance. Particularly, we define in symmetry two types of modal formulae, for knowledge and argumentation respectively, as well as two corresponding types of Kripke models. We then semantically establish the two-way generation method capable of transforming either type of Kripke model to the other type, and also rigidly justify our methodology by showing the syntactic invariance via a dual theorem. Our novel yet convincing approach to combine knowledge and argumentation fundamentally gets rid of any unnatural affectations in other previous attempts, while our simple yet clear framework is supposed to lay out a basic playground for both epistemic logic and argumentation theory.
Moreover, through careful examination on a vivid example of pragmatically utilizing our dual theory, we not only demonstrate in a concrete way how the methodology of two-way generation actually works, but we are also led to the eventual discovery of our framework's one significantly useful application in practice: realistically, the argument Kripke model is usually much easier to observe and obtain, for example simply by recording the actual verbal debate among the agents, from the source data of which we are then able to generate the corresponding epistemic Kripke model and carry out a real-time analysis over the current debate. What is more important, the above task essentially relies little on human intuition and so is also viable to get mechanically performed by an AI system, who can properly implement modal logic's decidability as one of its main advantages.
Further, we discuss possible expansions as well as applications of our framework, which may constitute some future research directions. Especially, we claim that in contrary to our common perception, the seemingly “static” knowledge essentially possesses an indispensably dynamic instinct. As Francis Bacon's famous saying goes: “Knowledge is power”. Such admitted characterization of knowledge also displays a vibrant image to help us intuitively understand knowledge's dynamic essence, as power can only manifest itself through actual exertion of the power in some specific dynamic event, for instance, by demonstrating your argument power to skillfully defeat your opponent during a debate, while your such power exactly comes out of your knowledge. All the entanglement between statics and dynamics reminds us of Karl Marx's extremely influential motto: “The philosophers have hitherto only interpreted the world in various ways; the point, however, is to change it”.

Acknowledgements
Both the authors owe immense gratitude to Teeradaj Racharak (whose nickname is `X') for his teaching and help on argumentation theory. We would also like to thank the anonymous reviewers for their invaluable advice.
